By John Milnor
Via John Milnor, from the Annals of arithmetic stories in Princeton college Press. Includes--Elementary proof approximately actual or advanced algebraic units, The curve choice lemma, The fibration theorem, and six extra chapters.
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Gauss, 1800]. 1981402347355922074... 13) Thus he obtained 19 accurate places in four iterations. Remark 81 In [G. Almkvist, B. Berndt, 1988] it is given another quantitative measure of the rapidity of convergence of the sequences (an )n≥1 and (bn )n≥1 . Define cn = a2n − b2n , n ≥ 0 and observe that cn+1 = an − b n c2n c2n = < . 2 4 · an+1 4 · M (a, b) Thus (cn )n≥0 tends to 0 quadratically. Remember that more generally, the convergence of the sequence (αn )n≥0 to L is of the mth order if there exist the constants C > 0 and m ≥ 1 such that |αn+1 − L| ≤ C · |αn − L|m , n ≥ 0 .
10 ), then M F i has two expressions, depending on the relation between M and Fi in the given point. For example, we have: M (a, b) [M (a, b) − F9 (a, b)] if F9 (a, b) ≤ M (a, b) . if F9 (a, b) ≥ M (a, b) M (a, b) − M (F 9) (a, b) = M 2 (a,b) 2M (a,b)−F9 (a,b) In [Silvia Toader, G. Toader, 2004, 2004a] is given the complete list of complementary means of a Greek mean with respect to another. Most of them are expressed using operations with the special means ∨ and ∧. Corollary 48 The complementaries of the Greek means are: G (A) = C3/2 , H(A) = C, C (A) = H , (A) = 1 ∨ + 3 ∧ − (∨ − ∧)2 + 4∧2 2 , (A) = 1 3 ∨ + ∧ − (∨ − ∧)2 + 4∨2 2 , F5 F6 (A) F7 (A) = F9 , F8 (A) F10 = = ∨2 + ∨ ∧ −∧2 (A) , F9 = F7 , 2 ∨ −∧ 1 2 ∨ + ∧ − ∧ (4 ∨ −3∧) .
They are related to the method of Archimedes for evaluation of π, Heron’s method of extracting square roots, Lagrange’s procedure of determination of the integral of some irrational functions and Gauss’ approximation of some elliptic integral. For the definition of these double sequences are used the arithmetic mean, the geometric mean and the harmonic mean. Then we define general double sequences using two arbitrary means. We study conditions on these means that assure the convergence of the sequences to a common limit and present methods for the determination of this limit.